When I first heard this quote I thought gee, that's really true. In my experience it happens all the time; I have some foggy notion of something and when I start to write it down I start to see the gaps in my initial logic and assumptions, paving the way for a much better idea. Eventually I can evolve it into a much stronger piece.
It's a great habit to just write things down, as they say. When I think about why this bit of writerly wisdom seems to be true, it jumped out at me that this is rooted in a basic physical fact:
Creating a good idea is NP-Hard.
This basic bit of Computer Science theory means that solving real problems is hard (generally believed to take exponential time) but verifying their solutions is easy (doable in polynomial time).
Mozart composing a great symphony is immensely hard, but appreciating can be done by anybody. Generating a mathematical proof to a tough theorem is hard and takes years, but verifying the theorem can be done by any trained mathematician. The pattern of examples of our society enjoying this relationship between creator and consumer can be found in every creative endeavor. (Off-topic: part of why education is so incredibly important is because it vaccinates everyone against outright lies...)
So when you finally write something down, you make the solution available to your own verifier. You've probably already vaguely known this intuitively (that you tend to sniff out a shitty proposal much faster than you formulate a good one) but without knowing why it worked or without much discipline. So now I urge you to remember and leverage this fact regularly:
Next time you struggle with any problem, it's probably because you're thinking too hard. It's really hard to judge the million half-formed ideas when they're in your head, but when you can finally read them, the flaws are obvious and only then can you improve on them. Just write! Draw! Make 20 prototypes! Do anything to get it into a medium that you can lay your eyes on!